Variable step-size LMS adaptive filters for CDMA multiuser detection

FACTA UNIVERSITATIS (NIS)SER.: ELEC. ENERG. vol. 17, April 2004, 21-32

Variable Step-Size LMS Adaptive Filters for CDMAMultiuser Detection

Karen Egiazarian, Pauli Kuosmanen,and Radu Ciprian Bilcu

Abstract: Due to its simplicity the adaptive Least Mean Square (LMS) algorithm iswidely used in Code-Division Multiple access (CDMA) detectors. However its con-vergence speed is highly dependent on the eigenvalue spread of the input covariancematrix. For highly correlated inputs the LMS algorithm has a slow convergence whichrequire long training sequences and therefore low transmission speeds. Another draw-back of the LMS is the trade-off between convergence speed and steady-state errorsince both are controlled by the same parameter, the step-size. In order to eliminatethese drawbacks, the class of Variable Step-Size LMS (VSSLMS) algorithms was in-troduced. In this paper, we study the behavior of some algorithms belonging to theclass of VSSLMS for training based multiuser detection in a CDMA system. We showthat the proposed Complementary Pair Variable Step-Size LMS algorithms highly in-crease the speed of convergence while reducing the trade-off between the convergencespeed and the output error.

Keywords: Code-division multiple access, adaptive filters, least mean square, vari-able step-size.

1 Introduction

Code-Division Multiple-Access (CDMA) using the direct-sequence (DS) spread-spectrum signaling has gained increased interest for application in telecommuni-cation systems. Some of the main advantages of the DS/CDMA technique are:the ability of asynchronous operation, a better channel usage compared with other

Manuscript received February 3, 2004. An earlier version of this paper was presented at the 6thInt. Conf. on Telecommunications in Modern Satellite, Cable and Broadcasting Services, TELSIKS2003, October 1-3, 2003, 18 000 Nis, Serbia.

The authors are with the Institute of Signal Processing, Tampere University of Technology, Ko-rkeakoulunkatu 1, 33720, Tampere, Finland, (e-mail: [karen,pqo,bilcu]@cs.tut.fi).

21

22 K. Egiazarian, P. Kuosmanen and R. C. Bilcu:

techniques that allow a single user to be transmitted over the channel at a certaintime and their ability to operate in the presence of narrow band communicationsystems. When a given user is demodulated in a DS/CDMA system, two types ofinterferences must be minimized, namely the wide band multiple access interfer-ence (MAI) and the Narrow Band Interference (NBI), as well as, the channel noise.The MAI is caused by other spread spectrum users into the channel while the NBIinterference is caused by other conventional communication systems.

Among other demodulation techniques, the adaptive methods has been success-fully applied to reduce both the MAI and NBI interferences in DS/CDMA systems.When the spreading code and the channel parameters of the desired user are knownor can be estimated, the blind adaptive detectors can be easily used [1]- [7], whereasin absence of these information the trained based implementations are preferred [8]-[10].

In the case of the trained based systems a known training sequence is trans-mitted which is used to tune the coefficients of the adaptive filter before the actualdata is send. The well known adaptive algorithm used in both blind and trainingbased demodulators is the Least Mean Square algorithm which has the advantageof having a simple implementation and low computational complexity. However,the main disadvantages of the LMS algorithm are its slow convergence when oper-ate with highly correlated input signals and the tradeoff between the convergencespeed and the output error [12]. In order to reduce these disadvantages many ofits variants where introduced in the open literature, such as, the class of VariableStep-Size LMS (VSSLMS) algorithms [13]- [15].

In this paper, we analyze the behavior of different VSSLMS adaptive algo-rithms for the problem of multiuser detection in synchronous CDMA systems. Weshow, by means of simulations, that the Complementary Variable Step-Size LMS(CPVSLMS) adaptive algorithm proposed by the authors in [15] possess a fasterconvergence speed than other known algorithms, while reducing or eliminating thetradeoff between convergence speed and steady-state output error.

The paper is organized as follows: in Section 2 the DS/CDMA signal modelis summarized and the adaptive trained based single user detector using the LMSadaptive filter is shortly reviewed, in Section 3 some variable step-sizes adaptivealgorithms are described, the simulation results are presented in Section 4 and Sec-tion 5 concludes the paper.

2 Theoretical Considerations

For the sake of simplicity we consider a synchronous CDMA system in which anumber of K users transmit over a single-path time-invariant channel. The process-

Variable Step-Size LMS Adaptive Filters for CDMA Multiuser Detection 23

ing gain is denoted by N, the attenuation of each user data are denoted by ak andthe data symbols transmitted by all users are aligned in time. The received signalsampled at chip rate can be written in vector form as follows:

r�n �� SAd

�n �� v

�n �� (1)

where the jth column of S represents the received spreading code of the jth user,the vector d

�n �� d1

�n �� dK

�n �� T contains the data symbols transmitted by all

users at the time instant n, the N � 1 vector v is the sampled channel noise and theK � K matrix A is given by:

A �� a1 0 0 �� 0

0 a2 0 �� 0��0 0 0 �� aK

��

Assuming that the desired user is user 1, a block diagram of a trained baseddetector using the standard LMS adaptive algorithm is depicted in Fig. 1, wherer�n � is the input vector described in Eq. (1), h

�n �� h1

�n �� hN

�n �� T

is the N � 1vector containing the coefficients of the demodulator, d1

�n � is the known desired

sequence that is the same as the data sequence transmitted by the user 1 and e�n � is

the output error.

r(n)h(n)

e(n)

+−

y(n)

d (n)1

Fig. 1. Block diagram of an adaptive detector using the LMS algorithm.

The LMS adaptive algorithm used to train the coefficients of the adaptive filterh�n � can be described by the following steps:

1. Compute the output of the adaptive filter h�n � :

y�n �� hT � n � r � n � � N

∑i ! 1

hi�n � ri

�n �� (2)

where ri�n � is the ith element of the vector r

�n � in Eq. (1).


2. Compute the output error:

e�n � � d1

�n ��" y

�n �� (3)

3. Update the coefficients of the adaptive demodulator:

h�n � 1 �#� h

�n �� µe

�n � r � n �� (4)

where µ is a constant parameter called step-size, which controlls the steady-state error and the convergence speed.

For the training based detector the convergence speed is governed by the eigen-value spread of the input autocorrelation matrix defined as:

R � E � r � n � rT � n � � � E � SAd�n � dT � n � AT ST � � E � v � n � vT � n � � � (5)

where E %$& represents the expected value of the quantity inside the squared bracketsand we have assumed that the elements of the vector v are random zero-mean andindependent from S, A and d

�n � .

It is clear from Eq. (5) that the eigenvalue spread of the input autocorrelationmatrix R can be far from unity and an adaptive demodulator using the standardLMS algorithm will have a very slow convergence. Since in the case of trainingbased detectors, during the adaptation period no data sequences can be transmit-ted this slow convergence will decrease also the transmission rate. Therefore, inpractical applications, the convergence speed of the detector has to be increasedwhile maintaining a small steady-state error. Besides a slow convergence, the stan-dard LMS algorithm has also the disadvantage of a trade-off between speed andsteady-state output error. Indeed from Eq. (4) we can see that in order to obtain asmall steady-state error, one has to chose a small step-size, but a small value of µdecreases the speed of convergence of the algorithm.

3 Variable Step-Size Least Mean Square Algorithm

In order to improve the performances of the LMS algorithm, the class of VSSLMSalgorithms was introduced. In this paper, we analyze and compare the behaviorof three algorithms belonging to this class. The comparison is done in terms ofconvergence speed, computational complexity and memory load.

The first compared algorithm is the Variable Step-Size LMS algorithm intro-duced in [13]. This algorithm uses a time-variable step-size in Eq. (4) which isadjusted as follows:

µ ' � n � 1 �#� αµ�n �(� γe2 � n �� (6)


where α and γ are two positive constant parameters, e�n � is the output error at time

instant n and

µ�n � 1 �#�*)+-, µmax if µ ' � n � 1 ��. µmax

µmin if µ ' � n � 1 ��/ µminµ ' � n � 1 � otherwise

(7)

µmin and µmax being the minimum and respectively the maximum values allowedfor the step-size.

In [13] the transient and steady-state analysis of the VSSLMS is given andthe theoretical missadjustment is derived for both stationary and non-stationarycases. However, from the analysis presented in [13] the value of the missadjustmentand the convergence speed depend on both coefficients α and γ . Therefore, wecan conclude that the VSSLMS increases the convergence speed but still has thedrawback between a fast convergence and a small steady-state error.

Another adaptive algorithm with time-varying step-size was introduced in [14]in order to improve the performances of the VSSLMS algorithm from [13] at thesteady-state. The step-size update of the Robust Variable Step-Size LMS algorithmof [14] is described by the following equations:

µ ' � n � 1 �#� αµ�n �(� γ p2 � n �� (8)

µ�n � 1 �#�*)+ , µmax if µ ' � n � 1 ��. µmax

µmin if µ ' � n � 1 ��/ µminµ ' � n � 1 � otherwise

(9)

where

p�n �0� β p

�n " 1 �(� �

1 " β � e � n � e � n " 1 �� (10)

Using an approximation of the error autocorrelation p�n � in the step-size up-

date, the influence of the measurement noise is reduced and the algorithm performsbetter at the steady-state. However, also in the case of this algorithm the steady-state missadjustment depend on all three parameters (α , β and γ), so the depen-dence between the convergence speed and the steady-state error still exist.

In order to eliminate the dependence between the convergence speed and thesteady-state error and in the same time to speed up the convergence, a new algo-rithm was introduced by the authors [15]. The new algorithm called Complemen-tary Pair Variable Step-Size LMS uses two adaptive filters that works in parallel asshown in Fig. 2. In this figure r

�n � is the N � 1 input vector that is common for


both adaptive filters, h1�n � and h2

�n � are two N � 1 vectors containing the coeffi-

cients of the accuracy and speed mode filter respectively, e1�n � , e2

�n � , y1

�n � and

y2�n � are the output errors and the output sequences of the adaptive filters and d1

�n �

is the known transmitted sequence of the first user (we assume that the user 1 is theuser of interest).

1h (n)

h (n)2

r(n) − +

− +

y (n)1

y (n)2

2

e (n)

e (n)

1

d (n)1

Fig. 2. Block diagram of the CPVSLMS.

The speed mode filter h2�n � uses an adaptive LMS algorithm with fixed and

large step-size µ2 � µmax, while the filter h1�n � uses a variable step-size µ1

�n � .

The speed mode filter is used in our algorithm just to speed-up the convergence.The accuracy mode filter h1

�n � is the filter of interest and from its output sequence

the transmitted data is reconstructed. The CPVSLMS algorithm is described by thefollowing steps:

1. Compute the output sequences and the output errors:

y1�n �� h1

�n � r � n ��

y2�n �� h2

�n � r � n ��

e1�n �� d1

�n ��" y1

�n ��

e2�n �� d1

�n ��" y2

�n � (11)

2. Update the coefficients of the speed mode filter:

h2�n � 1 �0� h2

�n �(� µmaxe2

�n � r � n �� (12)

3. Update the coefficients of the accuracy mode filter:

h1�n � 1 �1� )222+ 222

, h2�n � 1 �� if )+-, n

∑i ! n 3 T 4 1

e22�i ��/ n

∑i ! n 3 T 4 1

e21�i �

and n � T � 2T ��h1�n �(� µ1

�n � e1

�n � X � n �� otherwise

(13)


4. Update the step-size of the accuracy mode filter:

µ1�n � 1 �#�

)222222222+ 222222222,

µ1�n �� µmax

2� if )+ , n

∑i ! n 3 T 4 1

e22�i ��/ n

∑i ! n 3 T 4 1

e21�i �

and n � T � 2T ��max 5 αµ1

�n �� µmin 6 � if )+ , n

∑i ! n 3 T 4 1

e22�i ��7 n

∑i ! n 3 T 4 1

e21�i �

and n � T � 2T ��µ1�n �� otherwise

(14)

where T and α 89 0 � 1 are two constant parameters.The behavior of the proposed CPVSLMS algorithm can be described as fol-

lows: during the training period the speed mode filter performs similar as an adap-tive filter with fixed and large step-size. The accuracy mode filter performs also asan adaptive filter with fixed step-size for a number of T consecutive iterations (testinterval of length T). At the end of the test interval, the sum of squared errors of thespeed mode filter and accuracy mode filter are computed. If the sum of the squarederror of the accuracy mode filter is larger than the sum of squared error of the speedmode filter, it means that the speed mode filter performed better that the accuracymode filter during the last T iterations and it is closer to the Wiener solution. So,in this case the coefficients of the accuracy mode filter are updated with the coeffi-cients of the speed mode filter (because they are closer to the Wiener solution) andalso the step-size is increased (since a larger step-size had a better behavior). Thissituation appears at early stages of the adaptation when both adaptive filters arefar from the optimum. When the speed mode filter is near its steady-state its sumof squared error will be larger that the sum of squared error of the accuracy modefilter (larger step-size means larger steady-state MSE) and in this case the step-sizeof the accuracy mode filter is decreased in order to obtain the desired steady-statemissadjustment. The value of the step-size µ1

�n � at the steady-state will be equal

(or very close) to µmin.This operation of the CPVSLMS increases the convergence speed by increasing

the step-size µ1�n � when the adaptive filters are far from the optimum. Also the

trade-off between the steady-state error and convergence speed is eliminated sincethe missadjustment of the accuracy mode filter will be given only by the value ofµmin. In the case of CPVSLMS algorithm there are some fixed parameters that hasto be chosen by the user. The first parameter that controls the adaptation of the step-size and the convergence speed is α . For a small value of α , the step-size µ1

�n �

is decreased too fast and the convergence of the algorithm is decreased. Therefore,


we have used α � 0 � 9 that gives good results in all our experiments. The lengthof the test interval T also controls the convergence speed of the algorithm. If Tis chosen to be too large then the adaptation of the step-size of the accuracy modefilter is lost. For very large values of T the speed mode filter might converge insidethe test interval and the step-size µ1

�n � is not enough increased. In this case, the

speed of convergence will be very low. If T is too small, the step-size µ1�n � will

have large variations at the steady-state (even at the steady-state for a very smallnumber of consecutive iterations the speed mode filter might have smaller error).In all our experiments we have used T � 50 and we have obtained good results.The parameter µmax, must be chosen close to the stability limit in order to have afast convergence. The steady-state value of µ1

�n � will be equal with µmin (or very

close to it) so µmin will control the level of the steady-state missadjustment.

Table 1. Computational complexity and memory load for the compared algorithms.

Algorithm Mult. and Div. Add. and Sub. Memory load

LMS 2N+1 2N 2N+4

CPVSLMS 4N+5 4N+5 3N+10

VSSLMS 2N+4 2N+1 2N+6

RVSSLMS 2N+9 2N+3 2N+9

0 0.5 1 1.5 2x 104

−20

−15

−10

−5

0

5

Iterations

MSE

(dB

)

Mean Square Error for LMS with µmin

Fig. 3. Output Mean Square Error for the LMSwith µmin : 3 ; 10 < 4 .

0 0.5 1 1.5 2x 104

−20

−15

−10

−5

0

5

Iterations

MSE

(dB

)

Mean Square Error for LMS with µmax

Fig. 4. Output Mean Square Error for the LMSwith µmax : 3 ; 10 < 3 .

In Table 1 the computational complexity and memory load (the number ofmemory locations needed to store the variables and the parameters) of the com-pared algorithms are given. From this table we can see that the computationalcomplexity and memory load of the proposed CPVSLMS algorithm are almost


double compared with the other algorithms. However the benefit of the proposedalgorithm the increased convergence speed and the fact that the dependence be-tween the speed of convergence and the steady-state error is eliminated. Indeed thesteady-state missadjustment of the CPVSLMS is given by its steady-state step-sizewhich is µ1

�∞ �=� µmin whereas the speed of convergence can be tuned by chosen

the other parameters such as α , µmax and T . In the case of VSSLMS and RVSSLMSthe equations that gives the values of the parameters, provided in [13] and [14] aresometimes difficult to be used.

Table 2. Mean Square Error and the parameters for the compared algorithms.

Algorithm Parameters Steady-State MSE (dB)

LMS µmin : 3 ; 10 < 4 > 17 ? 4256LMS µmax : 3 ; 10 < 3 > 14 ? 0252

CPVSLMS µmin : 3 ; 10 < 4 @ µ2 : 3 ; 10 < 3 @ α : 0 ? 9 @T : 50

> 17 ? 4251

VSSLMS µmin : 3 ; 10 < 4 @ µmax : 3 ; 10 < 3 @ α : 0 ? 9 @γ : 0 ? 002

> 17 ? 2241

RVSSLMS µmin : 3 ; 10 < 4 @ µmax : 3 ; 10 < 3 @ α : 0 ? 97 @β : 0 ? 99 γ : 2

> 17 ? 2643

4 Simulations and Results

In this section, computer simulations showing the performances of theCPVSLMS algorithm are presented. The CPVSLMS is compared with the LMSwith fixed step-size, VSSLMS of [13] and MVSS of [14] in CDMA multiuser de-tection framework.

The signal model is given in Eq. (1), the number of users was K � 4 with thefirst user being the user of interest. The attenuation of the first user was 10 dB belowthe attenuation of the other three users. The spreading codes were chosen from aset of Gold sequences of length N � 31, the channel noise v

�n � was white Gaussian

with zero mean and variance σ 2v � 10 3 2. The transmitted data (the elements of the

vector d�n � in (1)) were equiprobable bipolar sequences with values in A(" 1 �� 1 B .

The parameters of all the tested algorithms are presented in Table 2 togetherwith the corresponding values of the steady-state missadjustments. The learningcurves (the value of the Mean Square Error during the adaptation) for all algorithmsare presented in Fig. 3, Fig. 4, Fig. 5, Fig. 6 and Fig. 7. These resultswere obtained by averaging a number of 100 Monte-Carlo simulations of length4 � 104 iterations. From these figures, we can see, that the CPVSLMS has fasterconvergence compared with the VSSLMS, RVSSLMS and the LMS having a smallstep-size while their steady-state MSE were comparable.


0 500 1000 1500 2000 2500 3000 3500 4000−20

−15

−10

−5

0

5

Iterations

MSE

(dB

)

Step−Size behaviour of the CPVSLMS

Fig. 5. Output Mean Square Error forCPVSSLMS.

0 500 1000 1500 2000 2500 3000 3500 4000−20

−15

−10

−5

0

5

Iterations

MSE

(dB

)

Step−Size behaviour of the VSSLMS

Fig. 6. Output Mean Square Error for VSSLMS.

0 500 1000 1500 2000 2500 3000 3500 4000−20

−15

−10

−5

0

5

10

Iterations

MSE

(dB

)

Step−Size behaviour of the RVSSLMS

Fig. 7. Output Mean Square Error for RVSSLMS.

0 0.5 1 1.5 2x 104

2

4

6

8

10

12

14

16x 10−4

Iterations

E{µ

1(i)}

Step−Size behaviour of the CPVSLMS

Fig. 8. Step-Size behavior for CPVSSLMS.

In order to have a more clear insight of the behavior of the compared algorithmsin Fig. 8, Fig. 9 and Fig. 10 the expected value of the step-size during theadaptation for the CPVSLMS, VSSLMS and RVSSLMS respectively are plotted.From these figures, we can see, that the step-size of the CPVSLMS algorithm hasthe smallest variations at the steady-state and also its value is very close to µmin �3 � 10 3 4. These results proofs our theoretical considerations that the steady-statemissadjustment of the CPVSLMS is given by µmin.

5 Conclusion

In this paper, we have compared the behavior of three different Variable Step-SizeLMS adaptive algorithms for the problem of multiuser detection in DS/CDMA


0 0.5 1 1.5 2x 104

0

0.5

1

1.5

2

2.5

3

3.5x 10−3

Iterations

E{µ

(i)}

Step−Size behaviour of the VSSLMS

Fig. 9. Step-Size behavior for VSSLMS.

0 0.5 1 1.5 2x 104

0

1

2

3x 10−3

Iterations

E{µ

(i)}

Step−Size behaviour of the RVSSLMS

Fig. 10. Step-Size behavior for RVSSLMS.

systems. All the compared algorithms uses a time-variable step-size adapted bythe output error to increase the adaptation speed. While two of the algorithmshave smaller computational complexity and memory load they still suffer from thefact that their steady-state error and the speed of convergence depend on the sameparameters. More that that, the equations used to compute the parameters of theVSSLMS and RVSSLMS are sometimes difficult to be used. The ComplementaryPair Variable Step-Size LMS algorithm introduced by the authors in [15], althoughhas an increased computational complexity and memory load, it has better speedperformance and more simple parameters setup which are very important in prac-tical applications.

References

[1] H. V. Poor and X. Wang, “Code-Aided Interference Suppression for DS/CDMACommunications-Part I: Interference Suppression Capability,” IEEE Trans. on Com-munications, vol. 45, no. 9, pp. 1101–1111, Sept. 1997.

[2] ——, “Code-Aided Interference Suppression for DS/CDMA Communications-PartII: Parallel Blind Adaptive Implementations,” IEEE transactions on Communica-tions, vol. 45, no. 9, pp. 1112–1122, Sept. 1997.

[3] Y. Gong, T. J. Lim, and B. F. Boroujeny, “Performance Analysis of the LMS BlindMinimum-Output-Energy CDMA Detector,” in Proc. of the IEEE International Con-ference on Acoustics Speech and Signal Processing, vol. 5, Istanbul, Turkey, 2000,pp. 2473–2476.

[4] S. V. M. Honig, U. M, “Blind Adaptive Multiuser Detection,” IEEE Trans. on Infor-mation Theory, vol. 41, no. 4, July 1995.


[5] K. W. Wong and T. O’Farell, “Blind Adaptive Single User Detection with AdjustableStep Size for DS/CDMA Communication over a Rayleyght Fading Mobile Chan-nel,” in Proc. of IEE Third International Conference on 3G Mobile CommunicationTechnologies, May May 8-10, 2002, pp. 180–184.

[6] V. Krishnamurthy and G. Yin, “Adaptive Step-Size Algorithms for Blind InterferenceSuppression in DS/CDMA Systems,” Trans. on Signal Processing, vol. 49, no. 1, Jan.2001.

[7] Y. Gong, B. F. Boroujeny, and T. J. Lim, “Variable Step-Size LMS Blind CDMA Mul-tiuser Detector,” in Proc. of the IEEE International Conference on Acoustics Speechand Signal Processing, vol. 4, Salt Lake City, 2001, pp. 2285–2288.

[8] M. Latva-aho and M. J. Juntti, “LMMSE Detection for DS-CDMA Systems in FadingChannels,” IEEE Trans. on Communications, vol. 48, no. 2, pp. 194–199, Feb. 2000.

[9] T. J. Lim, Y. Gong, and B. Farhang-Boroujeny, “Convergence Analysis of LMS Mul-tiuser CDMA Detectors,” in Proc. of Wireless Communications and Networking Con-ference, New Orleans, USA, 1999, pp. 75–79.

[10] S. L. Miller, “Training Analysis of Adaptive Interference Suppression for Direct-Sequence Code-Division Multiple-Access Systems,” IEEE Trans. on Communica-tions, vol. 44, no. 4, pp. 488–495, Apr. 1996.

[11] S. Verdu, Multiuser Detection. New York: Cambridge University Press, 1998.

[12] S. Haykin, Adaptive Filter Theory, 3-rd edition. New Jersey: Prentice-Hall, 1996.

[13] R. H. Kwong and E. W. Johnston, “A Variable Step Size LMS Algorithm,” IEEETrans. on Signal Processing, vol. 40, no. 7, pp. 1633–1642, July 1992.

[14] T. Aboulnasr and K. Mayyas, “A Robust Variable Step-Size LMS-Type Algorithm:Analysis and Simulations,” IEEE Trans. on Signal Processing, no. 3, pp. 631–639,Mar. 1997.

[15] R. C. Bilcu, P. Kuosmanen, and C. Rusu, “Improving Performances of Complemen-tary Pair LMS Algorithm,” in Proc. of X European Signal Processing Conference,EUSIPCO 2000, Tampere, Finland, Sept. 4–8, 2000.

Variable step-size LMS adaptive filters for CDMA multiuser detection

Documents

Transcript of Variable step-size LMS adaptive filters for CDMA multiuser detection